Arclength Consider a circle of radius r and an angle of x degrees as shown in the figure below. The segment of the circle opposite the angle x is called the arc subtended by x. We need a formula for its length. It is easily obtained using a "proportion" argument: r arc x If x = then the arc subtended by x is a semicircle which we know has length r. Now we argue that: A angle subtends an arc of length It follows that a 1 angle subtends an arc of length r ( r ) Therefore a angle subtends an arc of length = r ( r ) Likewise, a 3 angle subtends an arc of length 3 = 3r ( r ) In general, an angle of x degrees subtends an arc of length x r = xr We record this as a special result: In a circle of radius r, an angle of x degrees subtends an arc of length Example 1 In a circle of radius 5 inches, an angle of 9 subtends an arc of length This is close to 40.1 inches. xr (9) (5) inches. Example In a circle of radius 4.9 meters, an angle of 148 subtends an arc of length This is close to 1.7 meters. Example 3 In a circle of radius 5.87 feet, an angle of 5 36 48 subtends an arc of (148) (4.9) meters. ( 5 + 36 60 + 48 3600) (5.87) feet. Rounded to decimal places, the length is.6 feet. Note that we had to convert 5 36 48 into degrees because the formula gives the length when the angle is in degrees. Exercise 4 Calculate the length of the arc subtended by the given angle x on a circle with the given radius r. x = 115.3, r = 5 cm x = 4 48, r = 108 feet x = 3 1 45, r = 3960 miles 1
Practice problems 1. An angle of 37 subtends an arc of length 3 cm in some circle. What is the radius of the circle?. What angle, in degrees, minutes and seconds, (rounded to the nearest second), subtends an arc of length 40 feet in a circle of radius 18 feet? 3. What is the angle that the minutes hand of a clock turns through between 8:15 am and 8:57 am? 4. Find the length of the arc subtended by an angle of 175 4 45 in a circle of radius 8 yards. 5. What is the angle that the hour hand of a clock turns through between 7:00 am and 9:40 am.? 6. A clock has a minute hand that is.4 feet long. How many feet does its tip move between 9:43 am and 11:1 am? 7. (This is an old tricky problem.) At 1:00 noon, the hour hand and the minute hand of a clock point at the same spot. They NEXT point at the same spot, (after the minute hand has gone around once), some time after 1:05 pm. What is the exact time? 8. The figure shows a Norman window. The section WXY is a semi-circle and the section YUVW is part of a rectangle. If YU has length 3.5 feet and UV has length 3 feet, what is the total distance around the window? X Y W. U V 9. A running track consists of an inner rectangle of length 100 yards and width 50 yards, plus semicircular legs at both ends as shown in the figure below. There are 4 lanes and each lane is 1 yard wide. 50 yards 100 yards
(a) A runner does one lap in the inner-most lane. What is the total distance she covers? (You may assume that she runs along the innermost edge of her lane.) (b) What head-start should a runner in the outermost lane be given so that she covers the same distance, in one lap, as the runner in the inner-most lane? (You may also assume that she runs along the innermost edge of her lane.) Area of a sector The figure below shows a circle of radius r units and an angle of x degrees. The shaded part of the circle is called the sector subtended by the angle x. This time the question to address is: What is the area of the sector in terms of x and r? We follow the very route we took to find the length of a segment subtended by an angle of x degrees. The area of a circle with radius r is r square units. ( It follows that the area of the sector subtended by an angle of must be one half of r which is ) 1 r square units. In other ( words: An angle of degrees subtends a sector with area ) 1 r square units. Therefore: A 1 angle subtends a sector with area r 360 ( ) r A angle subtends a sector with area = r 360 360 ( ) r A 3 angle subtends a sector with area 3 = 3r 360 360 In general, an angle of x degrees subtends a sector with area ( ) r x = xr 360 360 For future use we record: In a circle of radius r units, an angle of x degrees subtends a sector with area xr 360 Example 5 In a circle of radius 16 cm, the sector subtended by an angle of 6 has area (16) 6 360 centimeters. This is approximately equal to 504. 9 square centimeters. square Exercise 6 You are given a circle of radius r and an angle θ. Calculate the area of the sector subtended by the angle. r =.5 cm, x = 14.3 r = 18 feet, x = 4 36 r = 3950 miles, x = 3 4 54 3
The formula is also simpler if the angle is measured in radians. It should be routine to conclude that: In a circle of radius r units, an angle of x radians subtends a sector with area xr Exercise 7 square units. 1. Find the area of the sector subtended by an angle of the given measure, in degrees, in a circle with the given radius. In each case, draw a diagram. Radius of circle Measure of angle Area of segment 5 inches 33 5 centimeters 3.7 4000 kilometers 0.14. A car window wiper blade consists of a metal holder 18 inches long to which a 15 inch rubber wiper is attached as shown in the figure below. If it rotates through 110, how much area does it wipe in one sweep? Rubber wiper 3. A vender sells two sizes of pizza by the slice. The small slice is 1 6 of a circular 18-inch-diameter pizza, and it sells for $.00. The large slice is 1 8 of a circular 6-inch-diameter pizza, and it sells for $3.00. Which slice provides more pizza per dollar? Angular Speed of a Rotating Object Angular and Linear Speeds Angular speeds arise where there are rotating objects like car tires, engine parts, computer discs, etc. When you drive a car, a gauge on the dashboard, (called a tachometer), continuously displays how fast your engine, (actually the crankshaft in your engine), is rotating. The number it displays at a particular time is called the angular speed of the engine at that instant, in revolutions per minute, (abbreviated to rpm). For example, an angular speed of 3000 rpm means that the engine rotates 3000 times in one minute. Since there isn t much space on the dashboard, 3000 rpm is actually displayed as 3 and you are instructed to multiply it by 1000, (there is symbol 1000 in the center of the tachometer). 4
For objects like the seconds hand of a clock, the minute hands of a clock, etc, that rotate relatively slowly, the angular speed is often given in degrees or radians per unit time. For example, the angular speed of the seconds hand of a clock may be given as 360 per minute, or 6 per second, (divide 3600 by 60), or 1600 per hour, (multiply 360 by 60). In general, to determine the angular speed of a rotating object per unit time do the following: (i) Fix a time interval, (ii) Record the amount of rotation in that time period, (iii) Divide the amount of turning by the length of the time interval. Exercise 8 1. What is the angular speed of the hour hand of a clock in: a. Degrees per hour b. Degrees per minute c. degrees second d. Revolutions per minute. Give the angular speed of the earth in: a. Degrees per hour b. Degrees per second c. degrees per minute Linear Speed of a Rotating Object The tires of a vehicle must rotate in order for the vehicle to move. When they rotate clockwise, it moves forward. It moves back when they rotate counter-clockwise. Take a typical car with tires that have radius 1 foot each. Imagine the tires making one full rotation clockwise. The figure below shows the position of a tire at the start, then after a half rotation and finally after a full rotation. R P R Q Q Q P 3.14 feet 3.14 feet R P At start After a half rotation After a full rotation (1) () The car has moved forward = 3.14 feet after the first half rotation. It moves another 3.14 feet after the second half rotation. Therefore it moves a total of 6.8 feet when the tire makes a full rotation. Complete the table. In case you have forgotten, 580 feet make a mile. Number of rotations Distance in feet moved by car 1 7 1. x 6.8 4 1.56 8 4 150 feet 1 mile To get an idea of a linear speed, assume that the tires rotate 6 times every second. Then the vehicle moves 6 () = 1 feet per second. This is close to 38 feet per second and it is called the linear speed of the tires, (and the vehicle) in feet per second. We may transform it into a speed per hour as follows: Since one hour is equal to 3600 seconds, the tires move 3600 1 feet in one hour. But 1 mile equals 580 feet. Therefore the linear speed of the tires is 3600 1 580 = 5.7 miles per hour 5
In general, the linear speed of a rotating circular object may be calculated as follows: Imagine a tire that has the same radius as the rotating circular object. Fix a time interval t and determine the distance s that the tire moves in that time. Now divide s by t. The result is the linear speed of the object. Example 9 A Farris wheel in an amusement park has radius 5 feet and it makes a full rotation every 1.6 minutes. What is the linear speed of point on the rim of the wheel in (i) feet per second, (ii) yards per minute? Solution: If the wheel were to roll along the ground, it would move 5 () feet in 1.6 minutes. (We have chosen a time interval of 1.6 minutes since we know how far it rolls in that time.) Therefore its linear speed is 50 feet per minute. 1.6 (i) Since 60 seconds make 1 minute, the linear speed in feet per second is ( ) 50 1.6 60 1.6 feet per second (ii) Since 3 feet make 1 yard, the linear speed in yards per minute is ( ) 50 1.6 3 3.7 yards per minute Exercise 10 1. The radius of the earth is about 3960 miles and there are 580 feet in a mile. The earth makes a full revolution in 4 hours. Calculate the linear speed of a point on the equator in: (a) Miles per hour. (b) Miles per minute. (c) Feet per second.. Each tire of a truck has radius 1.8 feet. Assume that they make 10 complete revolutions per second. Calculate: (a) The angular speed of the tires in degrees per second. (b) The linear speed of the truck in feet per second. (c) The linear speed of the truck in miles per hour. 3. Each tire of a certain car has radius 1. feet. At what angular speed, in revolutions per second, are the tires turning when it is moving at a constant speed of 70 miles per hour? (The angular speed of the tires may be different from the angular speed of the engine because of the gears.) 6
Angular Speed and Linear Speed Formulas The angular speed of an object that rotates at a constant rate is given by the formula Angular Speed = Angle swept out Time taken to sweep out the angle A conventional symbol for angular speed is the greek letter ω, (pronounced "omega"). Therefore if a rotating object sweeps out an angle of x degrees in some specified time t, (this may be in seconds, minutes, hours, or other units of time), then its angular speed is ω = x t degrees per unit time. Example 11 Wanda noticed that when the ceiling fan in her room is set to operate at "very low" speed, it makes a full rotation in 5 seconds. This means that a point of the fan sweeps out an angle of 360 in 5 seconds. Therefore its angular speed is ω = 360 5 = 7 degrees per second The linear speed of an object that is moving in a circle at a constant rate is the distance it travels per unit of time. To calculate it, one measures the distance s it travels along the circular path in a specified length of time t then divided s by t. A conventional symbol for speed is v. Therefore v = s t Example 1 Say a point at the end of a blade in the above fan is 6 inches from the center of rotation of the fan. Then in 5 seconds, such a point moves (6) = 5 inches. Therefore its linear speed is 5 5 inches per second. Rounded off to 1 decimal place, this is 3.7 inches per second. Relation Between Angular Speed and Linear Speed When a car is moving fast, its tires are rotating fast, and vice versa. Therefore angular speed and linear speed must be related by some equation. To determine such an equation, consider a car tire that is rotating. Say it has radius r feet and it rotates through x degrees in a period of t seconds. Then in those t seconds, the car travels a distance of rx feet. Therefore its linear speed is v = rx t feet per second. 7
Its angular speed is ω = x degrees per second t ( Notice that we may write the linear speed as v = r x ) ( x ) t feet per second. If we replace by ω we t conclude that the linear speed is v = rω feet per second. This is the required relation, if the angles are measured in degrees. Review Problems. 1. Convert 49 39 48 into decimal degrees and round off your answer to 3 decimal places.. Convert 50.53 into degrees, minutes and seconds and round off to the nearest second. 3. An angle of 48 subtends an arc of length 5 cm. in a given circle. What is the radius of the circle? 4. Calculate the area of the sector subtended by an angle of 48 in a circle of radius 5.8 feet. 5. An angle of 78.6 subtends a sector with area 14.5 square cm. in a given circle. What is the radius of the circle? 6. An object moves along a circle of radius 10 feet and it sweeps out an angle of 60 per second. Calculate its linear speed in (a) feet per second, (b) miles per hour. 7. How many inches does the tip of the minute hand of a clock move in 1 hour and 5 minutes if the hand is inches long? 8. Find the length of an arc subtended by an angle of 98 50 5 in a circle of radius 1 feet. 9. An object moves along a circle of radius 30 feet and it sweeps out an angle of 9 Calculate its linear speed in (a) meters per minute, (b) kilometers per hour. radians per minute. 10. Find the linear speed in miles per hour of the tire with radius 14 inches and rotating at 850 revolutions per minute. (The angular speed of a car tire may be different from the engine speed of the car because of gears.) 11. What is the area, rounded to 3 decimal places, swept out by a 9 inch minute hand of a clock between 8:10 am and 8:55 am? 1. There are two slices of pizza: one is 1 6 of a circular pizza with radius 14 inches and it costs $3.00. The other one is 1 5 of a circular pizza with radius 1 inches and it costs $.50. Which of the two provides more pizza per dollar? Show how you arrive at your answer. 8
Radian Measure of Angles A larger unit than a degree is called a radian. It is the angle that subtends an arc of length equal to the radius of a circle. It is shown in the figure below. Exercise 13 Draw an angle of: 1. radians by measuring off an arc of length r.. 3.5 radians by measuring off an arc of length 3.5r. 3. radians. 4..5 radians. To get the relation between degrees and angles, we read the above definition of a radian "backwards": In a circle of radius r, an arc of of length r is subtended by an angle of 1 radian It follows that In a circle of radius r, an arc of of length r is subtended by an angle of radians In a circle of radius r, an arc of of length 3r is subtended by an angle of 3 radians In a circle of radius r, an arc of of length r is subtended by an angle of radians Because the circumference of a circle with radius r is r, the sentence in bold letters tells us that the circumference of a circle subtends an angle of radians. But it also subtends an angle of 360. It follows that radians equal 360.Therefore This is a little over 57.3. 1 radian equals 360 = degrees We record it for future use: 1 radian = 360 = 57.3 degrees (1) 9
This conversion factor enables us to change from radians into degrees. For example, radians = = 360 degrees. 3 radians = 540 degrees 3.56 radians = 640. 8 x radians = x degrees degrees We record the formula for x radians as a special result: x radians are equal to x degrees. The common angles in radians (the equivalent measure in degrees are also included) are: Radians 0 6 4 3 3 3 4 5 6 Degrees 0 30 45 60 90 10 135 150 Radians 7 6 5 4 4 3 3 5 3 7 4 11 6 Degrees 10 5 40 70 300 315 330 360 We can also easily change from degrees into radians, by simply working backwards. More precisely, since degrees are equal to one radian, it follows that 1 degree equals radians. We also record this for later use: 1 degree = radian. () Using this, we conclude that: degrees = radians. 3 degrees = 3 radians. 4.7 degrees = 4.7 radians. x degrees = x radians. 10
Arclengths And Areas When Angles Are Measured In Radians One advantage a radian measure has over a degree measure is that we calculate arc-lengths and areas more easily when angles are measured in radians. To see how simple, note that in a circle of radius r units, an angle of 1 radian subtends an arc of length r units. It follows that: an angle of radian subtends an arc of length r units, an angle of 3 radian subtends an arc of length 3r units, an angle of 4.8 radian subtends an arc of length 4.8r units, an angle of x radians subtends an arc of length rx units. We record the last statement for future use: In a circle of radius r units an angle of x radians subtends an arc of length rx units Likewise in a circle of radius r, an angle of radians, (which is 360 ), subtends a sector with area r. It follows that: an angle of 1 radian subtends a sector with area r = r square units an angle of radians subtends a sector with area r square units an angle of 3.7 radians subtends a sector with area 3.7r square units an angle of x radians subtends a sector with area xr We also record the last statement for future use: square units In a circle of radius r units, an angle of x radians subtends a sector with area xr square units. Example 14 In a circle of radius 8 cm, an angle of.5 radians subtends an arc of length 8.5 = 70.0 cm. Example 15 In a circle of radius 3 ft, an arc of length 6.9 ft is subtended by an angle of 6.9 3 =.3 radians. Example 16 If an arc of length 66 meters is subtended by an angle of 4.8 radians then the circle must have radius 66 = 13. 75 meters. 4.8 Exercise 17 1. Convert each angle, (given in degrees), into radians. When necessary, round off your answer to 3 decimal places. 10 300 330 48 5 109 55.4 147 51 4. Convert each angle, (given in radians), into degrees. When necessary, round off your answer to decimal places.3 radians 3 7 radians 5.53 radians 15 radians 11 6 radians.1 radians 11
3. Find the length of the arc subtended by an angle of the given measure, in radians, in a circle with the given radius. In each case, draw a diagram. Radius of circle Measure of angle Length of arc 4.9 meters 3.77 radians 1.6 yards 4.8 radians 3000 kilometers 0.041 radians 1